modulus

The modulus of a complex number is its distance from the origin on the complex plane — how far the point a + bi sits from 0. It is the natural extension of absolute value: for a real number, absolute value measures distance from 0 on the line; for a complex number, modulus measures distance from 0 on the plane.

The modulus of z = a + bi is written |z| and computed by the Pythagorean theorem: |z| = sqrt(a^2 + b^2). This is always a non-negative real number, and it is zero only when z itself is zero. A neat link to the conjugate: |z|^2 = z times conj(z), since (a + bi)(a - bi) = a^2 + b^2.

A useful property is that the modulus respects multiplication: |z times w| = |z| times |w|. Addition is different, though — moduli only obey the triangle inequality |z + w| ≤ |z| + |w|, not equality, because two arrows can partly cancel.